Network Equations

3.6 Network Equations

For a network with $nb$ buses, all constant impedance elements of the model are incorporated into a complex $nb × nb$ bus admittance matrix $Ybus$ that relates the complex nodal current injections $I bus$ to the complex node voltages $V$ :

Ibus = YbusV.

(3.8)

Similarly, for a network with $nl$ branches, the $nl × nb$ system branch admittance matrices $Yf$ and $Yt$ relate the bus voltages to the $nl × 1$ vectors $If$ and $It$ of branch currents at the from and to ends of all branches, respectively:

If $[ ⋅ ]$ is used to denote an operator that takes an $n × 1$ vector and creates the corresponding $n × n$ diagonal matrix with the vector elements on the diagonal, these system admittance matrices can be formed as follows:

The current injections of (3.8)–(3.10) can be used to compute the corresponding complex power injections as functions of the complex bus voltages $V$ :

The nodal bus injections are then matched to the injections from loads and generators to form the AC nodal power balance equations, expressed as a function of the complex bus voltages and generator injections in complex matrix form as

gS(V,Sg) = Sbus(V ) + Sd − CgSg = 0.

(3.17)

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